Type HO String Theory
The Type HO String Theory also known as the Spin(32)/Z_2 Heterotic String Theory, or the SO(32) Heterotic String Theory, or the , is a 10-dimensional Heterotic String Theory. This means that the State is formed by tensoring the bosonic state (but with added Majarona-Weyl Fermions), with the Type II state. However, this raises the issue that the left-moving state is in 26 dimensional spacetime, but the right-moving state is in 10-dimensional spacetime. To fix these mismatched dimensions, we compactify the 16 mismatched dimensions on an even unimodular lattice, which means that the lattice has to have a Cartan Determinant of 1, and the vectors must have even magnitudes. For the Type HO String Theory, it is \frac{\operatorname{Spin}\left(32\right)}{\mathbb Z_2} . Action Principle The Action for the Type HO String Theory is of course the Heterotic String Action. It is given by the following action across the Worldsheet: Polchinski, Joseph. (1998). String Theory: Volume 2, p. = 45. S_H=\iint \frac{T}{2} \left( - \sum\limits_{k=1}^{32}{\left( i\hbar c_0 \bar{\lambda_{\mu}} \not\partial \lambda ^\mu \right)}-\left( i\hbar c_0\bar{\psi_{\mu }} \not\partial\psi^\mu \right) \right) \sqrt{-\det h_{\alpha\beta}}\mbox{ d}\sigma\mbox{ d}\tau The corresponding Lagrangian Density is obviously given by: : }_{\mathsf{\mathcal{H}}}}=\frac{T}{2} \left( - \sum\limits_{k=1}^{32}{\left( i\hbar c_0 \bar{\lambda_{\mu}} \not\partial \lambda ^\mu \right)}-\left( i\hbar c_0\bar{\psi_{\mu }} \not\partial\psi^\mu \right) \right) Where we removed the square root of the negative of the determinant of the Worldsheet metric because the Lagrangian Density is local at all points on the Worldsheet. Compactification of mismatched dimensions With this new action for the Type H String Theory being different from the ordinary RNS Action, we have solved the issue of the imbalance between bosons and fermions. Now, we need to tackle the problem of the left-movers existing in 26 dimensions and the right-movers existing in 10 dimensions, which seems like a serious inconsiswtencsy in the theory.. To do that, we will compactify the 26-10=16 mismatched dimensions between the left-movers and the right-movers, on a lattice, on some lattice.. Naturally, to preserve the symmetries of the Heterotic string action above, we need to make this lattice be Unimodular, or Self-Dual. The left-movers are from the Bosonic String State, so we have to consider compactifying 16 dimensions of the Bosonic String. So, we will do the following considerations: Compactification on this lattice with "dimensionless momenta" \vec v_L and \vec v_R would lead to the following condition: \|\vec v_L\|^2 - \|\vec v_R\|^2 + 2 (N - \tilde N) = 0 But since only the left-movers are Bosonic Strings and need to be compactified \vec v_R=0 ,. \|\vec v_L\|^2 = 0 - 2 (N - \tilde N)=-2\left(N-\tilde N\right) I.e. the norm-squared is even. A lattice made of such vectors is an even lattice, and thus, the lattice also needs to be somewhat even. The only suitable even, unimodular, 16 dimensional lattices are E(8)\times E(8) and \frac{\operatorname{Spin}\left(32\right)}{\mathbb Z_2} . The \mathrm{Spin} (32)/ \mathbb{Z}_2 String Theory is the latter. Note that \frac{\operatorname{Spin}\left(32\right)}{\mathbb{Z}_2} is also the gauge group of this String Theory. Unsuitability as a Theory of Everything The necessity for Heterotic Strings arose when it was found that Type IIB string theory was not suitable for the Theory of Everything and neither was Type IIA. The Type HO String Theory, however, is not suitable for the Theory of Everything either, because its gauge group cannot contain the Standard Model Gauge group as a Subgroup. However, the gauge group of Type HE is suitable for a Theory of Everything as E(8)\times E(8) can easily include the Standard Model Gauge group as a subgroup. ... T-Duality with Type HE String Theory Until the Second Superstring Revolution, it was thought that the two Heterotic String Theories were only connected due to their mismatched forms (i.e. with the 16 mismatched dimensions uncompactified.). However, this is useless, as this is neither a duality nor an equivalence, so one may not derive one String Theory from the other this way. During the Second Superstring Revolution, it was discovered that these two are actually related by T-Duality. The root lattice of E(8)\times E(8) is \Gamma^8\oplus\Gamma^8 , whereas the root lattice of \frac{\operatorname{Spin}\left(32\right)}{\mathbb Z_2} is \Gamma^{16} . Since: \Gamma^8\oplus\Gamma^8\oplus \Gamma^{1,1}=\Gamma^{16}\oplus \Gamma^{1,1} , The two types of Heterotic String Theory are T-Dual to each other. Category:String Theory